Generalized Least Squares
9.1 GLS Is More Efficient than OLS.
a. Equation (7.5) of Chap. 7 gives "ois = " + (X’X)1X’u so that E("ois) = " as long as X and u are uncorrelated and u has zero mean. Also,
var("ols) = E("ols – ")("ols – ")’ = E[(X, X)_1X, uu, X(X, X)_1]
= (X’X)1X’ E(uu’)X(X’X)1 = CT2(X, X)1X’fiX(X’X)1.
b. var("ols) – var("gls) = o2[(X’X)1X’fiX(X’X)1 – (X’fi1X)1]
= CT2[(X, X)1X, fiX(X, X)1 – (X’^1X)1X’^1fifi1 X(X’fi1X)1]
= ct2[(X’X)1X’ – (X’fi1X)1X’fi1]fi[X(X’X)1 – fi1X(X’fi1X)1]
= o2 AfiA’
where A = [(X’X)1X’ – (X’fi1X)1X’fi1]. The second equality post multiplies (X’fi1X)1 by (X’fi1X)(X’fi1X)1 which is an identity of dimension K. The third equality follows since the crossproduct terms give 2(X’fi1X)1. The difference in variances is positive semidefinite since fi is positive definite.
9.2 a. From Chap. 7, we know that s2 = e’e/(n – K) = u’PXu/(n – K) or
(n – K)s2 = u’PXu. Hence,
(n – K)E(s2) = E(u’PXu) = E[tr(u’PXu)]
= tr[E(uu’)PX ] = tr(SPX) = o2tr(fiPX)
and E(s2) = o2tr(fiPX)/(n – K) which in general is not equal to o2.
B. H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/97836425454819, © SpringerVerlag Berlin Heidelberg 2015
b. From part (a),
(n – K)E(s2) = tr(SPx) = tr(S) – tr(SPx)
but, both S and PX are nonnegative definite. Hence, tr(SPX) > 0 and (n — K)E(s2) < tr(S)
which upon rearranging yields E(s2) < tr(S)/(n — K). Also, S and PX are nonnegative definite. Hence, tr(SPX) > 0. Therefore, E(s2) > 0. This proves the bound derived by Dufour (1986):
0 < E(s2) < tr(S)/(n — K)
n
where tr(S) = o2. Under homoskedasticity o2 = o2 fori = 1,2, ..,n.
i=1
Hence, tr(S) = no2 and the upper bound becomes no2/(n — K). A useful bound for E(s2) has been derived by Sathe and Vinod (1974) and Neudecker (1977, 1978). This is given by 0 < mean of (n — K) smallest characteristic roots of S < E(s2) < mean of (n — K) largest characteristic roots of S < tr(S)/(n — K).
c. Using s2 = u0PXu/(n — K) = u0u/(n — K) — u0PXu/(n — K) we have
plim s2 = plim u0u/(n — K) — plim u0PXu/(n — K). By assumption plim u0u/n = o2. Hence, the first term tend in plim to o2 as n!1. The second term has expectation o2tr(PX^)/(n—K). But, PX^ has rank K and therefore exactly K nonzero characteristic roots each of which cannot exceed Xmax. This means that
E[u0PXu/(n — K)] < o2KXmax/(n — K).
Using the condition that Xmax/n! 0 as n!1 proves that lim E[u0PXu/(n — K)] ! 0
as n! 1. Hence, plim [u0PXu/(n — K)] ! 0 as n!1 and plim s2 = o2. Therefore, a sufficient condition for s2 to be consistent for o2 irrespective of X is that Xmax/n! 0 and plim (u0u/n) = o2 as n! 1, see Kramer and Berghoff (1991).
d. From (9.6), s*2 = e*0e*/(n – K) where e* = y* – X*"GLS = y* –
X*(X*,X*)_1X*,y* = PX*y* using (9.4), where PX* = In – PX* and PX* = X*(X*’X*)1X*’. Substituting y* from (9.3), we get e* = PX*u* where PX*X* = 0. Hence, (n — K)s*2 = e*0e* = u*’PX*u* with
(n – K)E(s*2) = E (u*’PX*u*) = E [tr (u*u*’PX*)]
= tr [E (u*u*0) PX* ] = tr (ct2Px*) = o2(n – K)
from the fact that var(u*) = o2In. Hence, E(s*2) = ct2 and s*2 is unbiased for o2.
9.3 The AR(1) Model.
= IT 
The multiplication is tedious but simple. The (1,1) element automatically gives (1 – p2). The (1,2) element gives – p + p(1 + p2) – pp2 = – p +
p + p3 — p3 = 0. The (2,2) element gives —p2 + (1 + p2) — pp = 1 — p2 and so on.
Again, the multiplication is simple but tedious. The (1,1) element gives
V і — pV і — p2 — p(—p) = (1 — p2) + p2 = 1, the (1,2) element gives ^ 1 — p2.0 — p.1 = —p, the (2,2) element gives 1 — p(—p) = 1 + p2 and so on.
c. From part (b) we verified that P1,P1 = (1 — p2)^1. Hence, £2/(1 — p2) = PP0 or £2 = (1 — p2)PP0. Therefore,
var(P1u) = P1var(u)P10 = o,2p1^P10
= CTu2(1 — p2)P1PP0P10 = ct82It
since o2 = cr62/(1 — p2).
9.4 Restricted GLS. From Chap. 7, restricted least squares is given by "rls = "ols + (X0X)1R0[R(X0X)1R0]1(r — R"ols). Applying the same analysis to the transformed model in (9.3) we get that "*ls = (X*0X*)1X*0y* = "GLS. From (9.4) and the above restricted estimator, we get
"RGLS = "GLS C (X*0X*)1R0[R(X*0X*)1R0]1(r — R"GLS)
where X* now replaces X. ButX*0X* = X0^1X, hence,
"RGLS = "GLS C (X0^1X)1R0[R(X0^1X)1R0]1(r — R"GLS).
9.5 Best Linear Unbiased Prediction. This is based on Goldberger (1962).
a. Consider linear predictors of the scalar yT+s given by yT+s = c0y. From (9.1) we getyT+s = c0X" C c0u and using the fact that yT+s = xT+s"Cux+s, we get
yT+s — yT+s = (c0X — x’x+s)" C c0u — ux+s.
The unbiased condition is given by E(yT+s — yT+s) = 0. Since E(u) = 0 and E(uT+s) = 0, this requires that c0X = xT+s for this to hold for every ". Therefore, an unbiased predictor will have prediction error
y T+s — yT+s = c’u — ut+s.
b. The prediction variance is given by
var (yT+s) = E (^t+s — Ут+s) (yT+s — Ут+s)0 = E(c0u — ux+s)(c0u — ux+s)0 = c0E(uu0)c C var(uT+s) — 2c0E(uT+su) = c0Ec C o. j:+s — 2c0m
using the definitions crT+s = var(uT+s) and m = E(uT+su).
c. Minimizing var(yT+s) subject to c0X = xT+s sets up the following Lagrangian function
§(c, X) = c0Ec — 2c0m — 2X0(X0c — xT+s)
where oT+s is fixed and where X denotes the Kx1 vector of Lagrangian multipliers. The first order conditions of § with respect to c and X yield a§/9c = 2£c — 2m — 2XX = 0 and Э§/ЭХ = 2X0c — 2xT+s = 0.
In matrix form, these two equations become
X0 о i
Using partitioned inverse matrix formulas one gets
£1[IT – X(X0£1X)1X0£1] £1X(X0£1X)1
(X0 £1X)1X0 £1 —(X0£1X)1
so that c = £1X(X0£1X)1xT+s + £1[IT — X(X0 £1X)1X0 £1]ш.
Therefore, the BLUP is given by yT+s = O0y = xT+s(X0£1X)1X0£1y + rn0£1y — rn0£1X(X0£1X)1X0£1y = xT+s" gls + m0£1y — m0£1X" gls = xT+s" gls + m0£1(y — X" gls)
= xT+s" GLS + ш0£ 1eGLS
where eGLS = y — X"GLS. For £ = a2^, this can also be written as y T+s = xT+s" GLS + m0^1eGLs/^2.
d. For the stationary AR(1) case
ut = put1 + є with ©t ~ IID (0, a2)
p <1 andvar(ut) = = a©2/(1 — p2). In this case, cov(ut, uts) = psa2
Therefore, for s periods ahead forecast, we get
^E(ut+su1)^ 
pT+s1 

E(ut+su) = 
E(uT+su2) 
= au2 
pT+s2 
^E(ut+sut) ) 
ps 
From £2 given in (9.9) we can deduce that ш = psc2 (last column of £2). But, £2_1£ = IT. Hence, £2_1 (last column of £2) = (last column of It) = (0, 0,.., 1/0. Substituting for the last column of £ the expression (ш/psc2) yields
£2_1 ш/psc2 = (0,0,.., 1)0
which can be transposed and rewritten as
ш’£1/с,2 = ps (0,0,.., 1).
Substituting this expression in the BLUP for yT+s in part (c) we get yT+s = xT+s"GLS + Ш0£_1eGLS = C2 = xT+s"GLS + ps(0, 0, .., 1)eGLS = xT+s3 GLS + pseT, GLS
where eT GLS is the Tth GLS residual. For s = 1, this gives y T+1 = xTC10 GLS + peT, GLS as shown in the text.
9.6 The W, LR and LM Inequality. From Eq. (9.27), the Wald statistic W can be interpreted as a LR statistic conditional on £, the unrestricted MLE of £, i. e., W = — 2log[maxL(B/£)/ maxL(B/£)]. But, from (9.34), we know
R"=r "
that the likelihood ratio statistic LR = — 2 log[maxL(", £)/maxL(", £)]. Using (9.33), maxU B/£) < maxL(B, £). The right hand side term is an
R"=r V ’ Rf=r,£
unconditional maximum over all £ whereas the left hand side is a conditional maximum based on £ under the null hypothesis Ho; R" = r. Also, from (9.32) maxL(B, £) = maxL(B/£). Therefore, W > LR. Similarly, from Eq. (9.31), the Lagrange Multiplier statistic can be interpreted as a LR statistic conditional on £, the restricted maximum likelihood of £, i. e., LM = —2 log[maxL("/£)/ maxL("/£)]. Using (9.33), maxL("/£) = maxL(", £)
R"=r " R"=r R"=r,£
and from (9.32), we get maxL("/£) < maxL(", £) because the latter is an
unconditional maximum over all £. Hence, LR > LM. Therefore, W > LR > LM.
9.7 The W, LR and LM for this simple regression with Ho; " = 0 were derived in problem 7.16 in Chap. 7. Here, we follow the alternative derivation proposed by Breusch (1979) and considered in problem 9.6. From (9.34), the LR is given
(<5, " = 0, 52) /L (dmle, "mle, &mle)
n
where ‘ = y, " = 0, 52 = (yi — y)2/nand
i=1
‘mle — ‘ols — y " olsX,
n n n
0 mle = 0 ols = E WE xi2, 5m le = E ei2/n i=1 i=1 i=1
and ei = yi — ‘ols — 0olsXi, see the solution to problem 7.16. But,
n
logL («,", 5^ =2 log 2k — 2 log 52 — У2 (Уі — ‘ — "Xi/2/252.
i=1
Therefore,
n
loggia, " = 0, 52) = —2 log 2 л — 2 log 52 — (yi — y)2/252
and
Therefore, lr=2— 2 log5 2+2 log 5m le = – log (5 2/5m le)
= nlog (TSS/RSS) = n log(1 /1 — R2)
where TSS = total sum of squares, and RSS = residual sum of squares for the simple regression. Of course, R2 = 1— (RSS/TSS).
Similarly, from (9.31) we have LM = —2 log maxL (a, "/52) / maxL(a, "/52)
But, maximization of L(a, "/52) gives aols and "ols. Therefore,
maxL(a, "/52) = L (a,", 52^
with
n
loggia,", 52) = —2 log 2 к — n log<52 — ^ e2/252.
i=1
Also, restricted maximization of L(a, "/(52) under Ho; " = 0 gives a = y and " = 0. Therefore, maxL(a, "/(52) = L(a, ", 52). From this, we conclude that
LM 2
Vi=1
= n — (Х)є2/Xy2 ) = n[1 — (RSS/TSS)] = nR2.
Finally, from (9.27), we have W=—2 log The maximization of L(a, "/52) gives ‘ols and "ols. Therefore,
maxL(a, "/cr2) = L(a,", 52).
a"
Also, restricted maximization of L(a,"/52) under " = 0 gives ca = y and " = 0. Therefore, maxL(a, "/(52) = L(a, " = 0, 52) with
n
logL(a," = a 52) = 22 log2 it — n log 52 — ^(yi — ^)2/2crmle.
2 2 i= 1
R2
RSS ) "VI – R2 This is exactly what we got in problem 7.16, but now from Breusch’s (1979) alternative derivation. From problem 9.6, we infer using this LR interpretation of all three statistics that W > LR > LM.
9.8 Sampling Distributions and Efficiency Comparison of OLS and GLS. This is based on Baltagi (1992).
2
a. From the model it is clear that ^ x2 = 5, yi = 2 + ui, y2 = 4 + u2, and
t=i
22 xtyt xtut
P ols = = = " + = = 2 + 0.2ui + 0.4u2
Let u0 = (u1, u2), then it is easy to verify that E(u) = 0 and £2 = var(u) =
The disturbances have zero mean, are heteroskedastic and serially correlated with a correlation coefficient p = —0.5.
b. Using the joint probability function P(u1, u2) and Pols from part (a), one gets
“Pols 
Probability 
1 
1/8 
1.4 
3/8 
2.6 
3/8 
3 
1/8 
Therefore, E(Pols) = " = 2 and var(Pols) = 0.52. These results can be also verified from Pols = 2 + 0.2u1 + 0.4u2. In fact, E(Pols) = 2 since
E(ui) = E(u2) = 0 and
var (" ois) = 0.04 var(ui) + 0.16 var(u2) + 0.16 cov(u15u2) = 0.04 + 0.64 – 0.16 = 0.52.
Also,
In fact, "GLS = (x0^ 1 x) Vfi 1y = 1/4(2y1 + y2) which can be
rewritten as " gls = 2 + 1/4[2u1 + u2]
Using P(u1,u2) and this equation for "GLS, one gets
" GLS 
Probability 
1 
1/8 
2 
3/4 
3 
1/8 
Therefore, E("GLS) = " = 2 and var("GLS) = 0.25. This can also be verified from "GLS = 2 + 1/4[2u1 + u2]. In fact, E("GLS) = 2 since E(u1) = E(u2) = 0 and
var ^" gls) = 16[4var(u1) + var(u2) + 4cov(u1,u2)] = 16[4 + 4 – 4] = 4.
This variance is approximately 48% of the variance of the OLS estimator.
c. The OLS predictions are given by yt = "olsxt which means that y 1 = "ols and y2 = 2"ols. The OLS residuals are given by et = yt — yt and their probability function is given by
(Єь e 2) 
Probability 
(0,0) 
1/4 
00 0 1 VO 
3/8 
(—1.6, 0.8) 
3/8 
("ols)] = 0.48 Ф var ok) = 0.52.
Similarly, the GLS predictions are given by yt = QGLSxt which means that yi = QGLS and y2 = 2QGLS. The GLS residuals are given by et = yt — yt and their probability function is given by
(Є1,Є2) 
Probability 
(0,0) 
1/4 
(1, —2) 
3/8 
(—1,2) 
3/8 
The MSE of the GLS regression is given by s2 = e0fi 1e = 1/3 [4e2 + 2e1e2 + e^] and this has a probability function
s2 
Probability 
0 
1/4 
4/3 
3/4 
with E(s2) = 1. An alternative solution of this problem is given by Im and Snow (1993).
9.9 Equicorrelation.
a. For the regression with equicorrelated disturbances, OLS is equivalent to GLS as long as there is a constant in the regression model. Note that
1 p p… p
fi = P 1 P ••• P
p p p… 1
so that ut is homoskedastic and has constant serial correlation. In fact, correlation (ut, ut_s) = p for t ф s. Therefore, this is called equicorrelated. Zyskind’s (1967) condition given in (9.8) yields
Px^ = fiPx.
In this case,
Px^ = (1 — p)Px + pPxtTtT
and
^Px = (1 — p)Px + ptTtTPx.
But, we know that X contains a constant, i. e., a column of ones denoted by tT. Therefore, using PXX = Xwe get PXtT = tT since tT is a column of X. Substituting this in PX^ we get
Px^ = (1 — p)Px + ptTtT.
Similarly, substituting tTPX = tT in £2PX we get £2PX = (1 — p)PX + ptTtT. Hence, £2PX = PXЙ andOLS is equivalent to GLS for this model.
b. We know that (T — K)s2 = u0PXu, see Chap. 7. Also that
E. u’I^u) = E[tr(uu0 PX)] = tr[E(uu0PX)] = tr(o2^PX)
= o2tr[(1 — p)Px + pltlT^P x] = ct2(1 — p)tr(Px)
since tTPx = tT — tTPX = tT — tT = 0 see part (a). But, tr(PX) = T — K, hence, E(u0PXu) = o2(1 — p)(T — K) andE(s2) = o2(1 — p).
Now for £2 to be positive semidefinite, it should be true that for every arbitrary nonzero vector a we have a0^a > 0. In particular, for a = tT, we get
tT^tT = (1 — p)tT tT + p tT tT tT tT = T(1 — p) + T2p.
This should be nonnegative for every p. Hence, (T2 — T)p + T > 0 which gives p > — 1/(T — 1). But, we know that p < 1. Hence, —1/(T — 1) <
p < 1 as required. This means that 0 < E(s2) < [T/(T — 1)]o2 where the lower and upper bounds for E(s2) are attained at p = 1 and p = —1/(T — 1), respectively. These bounds were derived by Dufour (1986).
9.10 a. The model can be written in vector form as: y = ain + u where y0 =
(yi,..,yn), in is a vector of ones of dimension n, and u0 = (ui,..,un). 1 n
Therefore, &ols = (i^in i^y = yi/n = y and
_p p.. 1_
where In is an identity matrix of dimension n and Jn is a matrix of ones of dimension n. Define En = In — Jn where Jn = Jn/n, one can rewrite S as S = o2[(1 — p)En + (1 + p(n — 1))Jn] = o2^ with
Therefore,
i0T i i n n[4] n 
‘gls = (ins 1in) 1ins 1y =
inJny
inJny = = y
nn
b. s2 = e0e/(n — 1) where e is the vector of OLS residuals with typical element ei = yi — y for i = 1,.., n. In vector form, e = Eny and
s2 = y0Eny/(n — 1) = u0Enu/(n — 1) since En in = 0. But,
E(u0Enu) = tr(SEn) = o2tr[(1 — p)En] = o2(1 — p)(n — 1)
since EnTn = 0 and tr(En) = (n — 1). Hence, E(s2) = o2(1 — p) and E(s2) — o2 = — po2.
This bias is negative if 0 < p < 1 and positive if — 1/(n — 1) < p < 0.
c. s2 = eGLS^_1eGLS/(n — 1/ = e,^_1e/(n — 1/ where eGLS denotes the vector of GLS residuals which in this case is identical to the OLS residuals. Substituting for e = Eny we get 2 y0En^_1 Eny u0En^_1Enu
s* = n — 1 = (n — 1/
E(s^) = o2tr(^En^ 1En//(n — 1/ = 
tr[En] = o2 
since Entn = 0. Hence,
d. truevar(‘ols/ = (i^n/ 1 i^n^in/ 1 = i^in/n2
= o2 [(1 + p(n — 1//inJnin]/n2 = o2[1 + p(n — 1/]/n (9.1)
which is equal to var (aGLS/ = (i^S1^) 1 as it should be. estimated var((aols/ = s2 (i^in) 1 = s2/nsothat
E[estimated var(aols/ — true var(aols/] = E(s2//n — o2[1 + p(n — 1/]/n
= o2[1 — p — 1 — p(n — 1/]/n = —po2.
9.15 Neighborhood Effects and Housing Demand
a. This replicates the first three columns of Table VII in Ioannides and Zabel (2003, p. 569) generating descriptive statistics on key variables, by year:
. by year, sort: sum price pincoml highschool changehand white npersons married
> year = 1985 Variable 
Obs 
Mean 
Std. Dev. 
Min 
Max 
price 
1947 
81.92058 
25.0474 
44.89161 
146.1314 
pincom1 
1947 
28.55038 
15.47855 
3.557706 
90.00319 
highschool 
1947 
.8361582 
.3702271 
0 
1 
changehand 
1947 
.2891628 
.4534901 
0 
1 
white 
1947 
.8798151 
.3252612 
0 
1 
npersons 
1947 
2.850539 
1.438622 
1 
11 
married 
1947 
.7134052 
.4522867 
0 
1 
> year = 1989 

Variable 
Obs 
Mean 
Std. Dev. 
Min 
Max 
price 
2318 
116.7232 
49.82718 
48.3513 
220.3118 
pincom1 
2318 
47.75942 
30.3148 
4.444763 
174.0451 
highschool 
2318 
.8597929 
.3472767 
0 
1 
changehand 
2318 
.3170837 
.4654407 
0 
1 
white 
2318 
.8658326 
.3409056 
0 
1 
npersons 
2318 
2.768335 
1.469969 
1 
11 
married 
2318 
.6535807 
.4759314 
0 
1 
> year = 1993 Variable 
Obs 
Mean 
Std. Dev. 
Min 
Max 
price 
2909 
115.8608 
44.73127 
53.93157 
240.2594 
pincom1 
2909 
50.07294 
29.95046 
6.201 
184.7133 
highschool 
2909 
.8697147 
.3366749 
0 
1 
changehand 
2909 
.2781024 
.4481412 
0 
1 
white 
2909 
.8480578 
.3590266 
0 
1 
npersons 
2909 
2.738398 
1.435682 
1 
9 
married 
2909 
.6452389 
.4785231 
0 
1 
This replicates the last column of Table VII in loannides and Zabel (2003, p.569) generating descriptive statistics on key variables for the pooled data:
. sum price pincom1 highschool changehand white npersons married
Variable 
Obs 
Mean 
Std. Dev. 
Min 
Max 
price 
7174 
106.9282 
44.90505 
44.89161 
240.2594 
pincom1 
7174 
43.48427 
28.45273 
3.557706 
184.7133 
highschool 
7174 
.8574017 
.3496871 
0 
1 
changehand 
7174 
.2936995 
.4554877 
0 
1 
white 
7174 
.8624198 
.3444828 
0 
1 
npersons 
7174 
2.778506 
1.448163 
1 
11 
married 
7174 
.6664343 
.4715195 
0 
1 
b. This replicates column 1 of Table VIII of Ioannides and Zabel(2003, p. 577) estimating mean of neighbors housing demand. The estimates are close but do not match.
. reg lnhdemm lnprice d89 d93 lnpincomem highschoolm changehandm whitem npersonsm marriedm hagem hage2m fullbathsm bedroomsm garagem
7174
788.35
0.0000
0.6066
0.6058
.28272
lnhdemm 
Coef. 
Std. Err. 
t 
P> t 
[95% Conf. Interval] 

lnprice 
.2429891 
.0266485 
9.12 
0.000 
.295228 
.1907503 
d89 
.0967419 
.0102894 
9.40 
0.000 
.1169122 
.0765715 
d93 
.1497614 
.0109956 
13.62 
0.000 
.1713159 
.1282068 
lnpincomem 
.3622927 
.0250064 
14.49 
0.000 
.3132728 
.4113126 
highschoolm 
.1185672 
.0263588 
4.50 
0.000 
.066896 
.1702383 
changehandm 
.0249327 
.0179043 
1.39 
0.164 
.0101651 
.0600305 
whitem 
.2402858 
.0144223 
16.66 
0.000 
.2120139 
.2685577 
npersonsm 
.0692484 
.0069556 
9.96 
0.000 
.0828834 
.0556134 
marriedm 
.1034179 
.0236629 
4.37 
0.000 
.0570315 
.1498042 
hagem 
.0074906 
.0009053 
8.27 
0.000 
.0057159 
.0092652 
hage2m 
.008222 
.0010102 
8.14 
0.000 
.0102023 
.0062417 
fullbathsm 
.2544969 
.0085027 
29.93 
0.000 
.2378291 
.2711647 
bedroomsm 
.1770101 
.009258 
19.12 
0.000 
.1588616 
.1951586 
garagem 
.2081956 
.0135873 
15.32 
0.000 
.1815604 
.2348308 
_cons 
2.861724 
.1188536 
24.08 
0.000 
2.628735 
3.094712 
This replicates column 2 of Table VIII of loannides and Zabel(2003, p.577) estimating a standard housing demand with no neighborhood effects. The estimates do not match. This may be because the authors included only one observation per cluster. Here, all observations are used.
. reg lnhdem lnprice lnpincome highschool changehand white npersons married d89 d93, vce(cluster neigh)
Linear regression Number of obs = 7174
F(9,364) = 24.53
Prob > F = 0.0000
Rsquared = 0.1456
Root MSE = .49666
(Std. Err. adjusted for 365 clusters in neigh)
Robust

This replicates column 3 of Table VIII of loannides and Zabel(2003, p.577) estimating a reduced form housing demand. The estimates do not match.
. reg lnhdem lnprice lnpincome highschool changehand white npersons married d89 > d93 lnpincomem highschoolm changehandm whitem npersonsm marriedm hagem hage2m fullbathsm bedroomsm garagem, vce(cluster neigh)
Linear regression 
Number of obs 
= 7174 
F(20, 364) 
= 34.27 

Prob > F 
= 0.0000 

Rsquared 
= 0.4220 

Root MSE 
= .40883 
(Std. Err. adjusted for 365 clusters in neigh)

References
Dufour, J. M. (1986), “Bias of s2 in Linear Regressions with Dependent Errors,” The American Statistician, 40: 284285.
Kramer, W. and S. Berghoff (1991), “Consistency of s2 in the Linear Regression Model with Correlated Errors,” Empirical Economics, 16: 375377.
Neudecker, H. (1977), “Bounds for the Bias of the Least Squares Estimator of s2 in Case of a FirstOrder Autoregressive Process (positive autocorrelation),” Econometrica, 45: 12571262.
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